The generation and subsequent recording of sonic waves is a key measurement employed in oil well logging. Traditionally, a key characteristic estimated from waveforms has been the formation compressional slowness (the reciprocal of velocity). Since for an uncased borehole the compressional head wave is the first arrival in the waveform, the compressional slowness can be estimated merely by measuring the first time of arrival of energy at two receivers located a known distance apart. A drawback of this method is that the first arrival detection time is prone to error in the presence of noise spikes. Also, in a cased borehole with poorly bonded cement, a strong casing wave usually arrives before the compressional head wave and tends to interfere with its detection.
In recent years attempts have been made to extract additional information from sonic waveforms, such as the compressional and shear waves, U.S. Pat. No. 4,210,966 to Ingram and the Stoneley wave U.S. Pat. No. 4,131,875 to Ingram. These patents describe a multiple fold cross-correlation technique to determine the slowness of these waves from a plurality of waveforms detected by an array of receivers.
Techniques have been employed and described wherein a sonic tool is used with a receiver array containing a large number of receivers, such as 12. See copending U.S. patent application Ser. No. 335,758 filed on Dec. 30, 1981 by Christopher V. Kimball and Thomas L. Marzetta entitled "Sonic Well Logging" and assigned to the same assignee as for the present invention.
Signal processing methods are needed to estimate the acoustic slownesses of interest from the recorded arrays of waveforms. As in other signal processing applications, there is always a fundamental tradeoff between resolution and measurement statistics (i.e., accuracy, or variance). In acoustic logging, as the array aperture gets longer, the quality of the slowness estimate increases while the resolution decreases. This occurs since the processing tends to average, or smooth, the actual slowness variation in the vicinity of the array over the length of the aperture. As the array aperture increases, more averaging occurs, and resolution is sacrificed for improved statistics. Conversely, a short aperture provides less averaging, gives a noisier estimate but is more responsive to rapid variations.
Besides the fundamental resolution-versus-statistics tradeoff, the wavelength at acoustic frequencies also plays a fundamental role in that it limits the resolving power of the method. As a rule of thumb, one cannot resolve variations in the slowness which occur over distances much less than a wavelength. An acoustic wavelength, assuming a sound speed of 6 km/s, at 10 kHz is roughly 2 feet. Slower sound speeds have proportionately shorter wavelengths. Accordingly, sonic logging methods may resolve beds on the order of a foot thick or more when the operating frequency is approximately 10 kHz.
Sonic array processing methods such as the slowness-time coherence (STC) process described in the foregoing Kimball et al patent application and the maximum likelihood method as described in an article entitled "High Resolution Phase Velocity Spectra Estimation" by A. Baggeroer et al and presented at pages 471, 472 of the International Conference On Digital Signal Processing held in Florence, Italy on Sept. 2-5, 1981 among others are distinctly different, but have two important features in common. Each processes the data associated with a given source shot independent of the data associated with all other several shots and each is based on the assumption that the formation is homogeneous across the receiver aperture. In practice the formation is unlikely to be homogeneous, particularly over long intervals as an aperture formed by a twelve receiver array using six inch spacing between receivers. Furthermore, if a shorter array is used to achieve a resolution of the order of a foot, the measurement tends to be noisy.
Techniques have been proposed to provide borehole compensated sonic logs. One such method is described in an article entitled "A New Approach To Sonic Logging And Other Acoustic Measurements" by Kokesh et al and published in the March, 1965 issue of the Journal of Petroleum Technology. In this technique a pair of sonic transmitters are used, one below and one above four receivers. Two interval transit time (slowness) measurements are made by different pairs of receivers for the same depth intervals but using sonic pulses generated by the different transmitters. The interval measurements are combined and averaged to obtain a slowness measurement which is compensated for different conditions such as cave-ins and tooltilts.